FINC3017 · Investments And Portfolio Management
Performance Evaluation: Skill vs Luck
Performance Evaluation: Skill vs Luck (Week 10) is the toolkit for judging a portfolio after the fact. You distinguish the time-weighted return (which strips out cash-flow timing to isolate manager skill) from the money-weighted return (the investor's actual IRR), then compute the risk-adjusted measures — Sharpe, M², Treynor, Jensen's alpha, the information ratio and Sortino — and, crucially, choose the right one for the question. Market-timing tests add a squared term to the regression, attribution splits excess return into allocation and selection, and survivorship bias warns that the dead funds are missing from the data.
What this chapter covers
- 01Time-weighted return (TWR) = ∏(1 + R_i) − 1 vs money-weighted return (MWR / IRR)
- 02Sharpe = (E[r_p] − rf)/σ_p; M² = rf + Sharpe·σ_M
- 03Treynor = (E[r_p] − rf)/β_p; Jensen α = E[r_p] − [rf + β_p(E[r_M] − rf)]
- 04Information ratio = α/σ(ε); Sortino = (E[r_p] − target)/downside deviation
- 05Choosing the metric: total vs systematic risk, vs model, vs benchmark, downside-only
- 06Market-timing tests: Treynor-Mazuy quadratic and Henriksson-Merton
- 07Attribution: allocation effect vs selection effect
- 08Skill-vs-luck: persistence, survivorship bias, value added = α × fund size
The performance-metric showdown
- 1 markSharpe ratio = (E[r] − rf)/σ = (14% − 3%)/22% = 11/22 = 0.50.
- 1 markTreynor measure = (E[r] − rf)/β = (14% − 3%)/1.3 = 11/1.3 = 8.46%.
- 2 marksJensen's alpha = E[r] − [rf + β(E[R_m] − rf)] = 14% − [3% + 1.3 × (11% − 3%)] = 14% − [3% + 10.4%] = 14% − 13.4% = +0.6%.
- 1 markInformation ratio = α/σ(ε) = 0.6%/5% = 0.12.
- 2 marksM² = rf + Sharpe·σ_m = 3% + 0.50 × 17% = 3% + 8.5% = 11.5%.
- 1 markVerdict: M² = 11.5% exceeds the 11% market return and Jensen's alpha is positive (+0.6%), so P outperformed on a risk-adjusted basis, though only modestly.
Key terms
- Time-weighted vs money-weighted return
- The time-weighted return chains sub-period returns, ∏(1 + R_i) − 1, removing the effect of when cash flowed in or out — it judges manager skill. The money-weighted return is the IRR that depends on cash-flow timing — it is the investor's actual experienced return.
- Treynor measure
- Excess return per unit of systematic risk, (E[r_p] − rf)/β_p. It is the right risk-adjusted measure when the portfolio is one component of a larger diversified holding, so only its market (beta) risk matters, not its total volatility.
- M-squared (M²)
- A risk-adjusted measure, M² = rf + Sharpe·σ_M, that re-expresses the Sharpe ranking in percentage-return units at the market's level of risk. It can be compared directly with the market return: an M² above the market return signals outperformance.
- Sortino ratio
- A variant of the Sharpe ratio that divides excess return over a target by downside deviation rather than total standard deviation. It penalises only harmful (below-target) volatility, making it appropriate when investors care about losses but not upside swings.
- Survivorship bias
- The upward distortion in measured performance that arises when failed or closed funds drop out of a database, leaving only survivors. It makes the average manager look more skilled than reality and is a key reason apparent persistence overstates true skill.
Performance Evaluation: Skill vs Luck FAQ
When do I use Sharpe versus Treynor?
Use the Sharpe ratio when the portfolio is the investor's entire holding, because total risk (σ) is what they bear. Use the Treynor measure when the portfolio is just one sleeve of a larger, well-diversified portfolio, so only its systematic (beta) risk is relevant. Both reward excess return, but they divide by different risk measures, which can rank funds differently.
Why does the time-weighted return judge skill better than the money-weighted return?
The money-weighted return (IRR) is sensitive to when investors add or withdraw cash — decisions the manager usually does not control. The time-weighted return chains the period returns and so neutralises those flows, isolating the performance of the manager's actual investment decisions. That is why performance league tables use TWR, while an individual investor's realised outcome is the MWR.
How do you test whether a manager can time the market?
Add a squared market-excess-return term to the characteristic-line regression — the Treynor-Mazuy model r_p − rf = α + β(r_m − rf) + γ(r_m − rf)² + ε. A positive and significant γ means the manager raises beta in up markets and lowers it in down markets, the signature of genuine timing skill. The Henriksson-Merton variant uses a down-market indicator instead of a square.
Exam move
Build the full metric panel for a described manager (Sharpe, Treynor, Jensen, IR, M²) quickly and accurately, and pair each number with when you would actually use it. Memorise the TWR-versus-MWR distinction and the Treynor-Mazuy timing test, because the exam loves both the computation and the which-metric reasoning.
